A critical challenge in image restoration is the presence of various types of noise. Meanwhile, noise detection is a crucial step in mixed noise removal. This paper tackles the challenge of restoring images corrupted by a mixture of additive Gaussian and multiplicative Gamma noise. In the proposed method, we integrate the noise detection process into a variational model using a dual formulation of a maximum a posteriori (MAP) estimator. The variational model consists of a novel adaptive fidelity term and a plugin-and-play regularization term. The fidelity term contains an adaptive weight that can automatically detect the noise types, levels, and pollution ways for each pixel. There is flexibility in choosing a plugin-and-play regularization term. For example, we can use a model-based regularizer or a deep learning-based regularizer. In addition, we present a splitting algorithm to minimize the proposed cost functional. This splitting technique enables us to transfer a mixed noise removing problem to several subproblems, including noise removal and detection. The noise detection process can be iteratively estimated by the proposed algorithm itself. Therefore, in the numerical experiments, the proposed model outperforms the existing Rudin-Osher-Fatemi (ROF), Aubert-Aujol (AA), BM3D, and deep learning-based single type denoiser. Experimental results show that the proposed model can remove noise more efficiently and better preserve details in images. Compared to the existing best-performing single type denoiser, on average, the improvements of PSNR values range from 0.33 dB to 0.81 dB under noise mixture ratios α=0.4,0.6 .